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Multiplication ... without knowing your tables!

The ancient Egyptians and present day Russians use a simple method of performing multiplication of long numbers without using a calculator and that does not depend on knowing your multiplication tables. All it needs is the ability to double a number and to halve it.

Here is an example:

What is 89 times 104 ? Put 89 and 104 at the top of two columns. The first column is filled by doubling the number above it and the second column's rule is to halve each number to get the next one below it in that column. Keep filling the rows until the halving column gets down to zero. If you get a remainder when halving (e.g by halving an odd number), just forget it but mark those odd numbers with a star(*):

 
    89     104
   178      52
   356      26
   712      13 *
  1424       6
  2848       3 *
  5696       1 *
  

To find the product of 89 and 104, add up those numbers in the first column where there is a star in the second column:

712 + 2848 + 5696 = 9256 = 89 x 104

The reason this works depends on writing numbers in the binary scale or writing numbers as sums of powers of 2. Although this system goes back to around 3000 BC and is one of the earliest known methods of multiplication, it is also exactly what computers do when working internally in binary.

Can you... verify the process works if we worked out 104 times 89, so that this time we are halving in the 104 column and doubling in the 89 column? Try it on 32 times 123. This example is very easy since there is no addition to do! Why?